How to keep the original shape when simulating cloth?
Try increasing Bending and Structural. To have an order of magnitude, you can check the Presets :


Silk : Bending 0.05 ; Structural 5.00
Rubber : Bending 25.00 ; Structural 15.00


Stiffness Scaling allows to control the stiffness of specific parts of the mesh. Create two vertex group (one for Struct, one for Bending), assign all your vertices to both groups, them weight-paint them.


Blue vertices will have the stiffness defined in Overall stiffness
Red vertices will have a stiffness of Overall + Max